Discretization of parameter identification in PDEs using Neural Networks

We consider the ill-posed inverse problem of identifying parameters in a time-dependent PDE model, whose nonlinearity is supposed to be unknown. The model nonlinearity is represented via a neural network; suggesting an all-at-once approach, we bypass the need for training data. In the general case, the approximation via a neural network can be realized as a discretization scheme. Therefore, we study discretization of regularization in terms of Tikhonov and Landweber methods for the inverse problem, and prove convergence when the discretization error and noise level tend to zero.